Second-order Fusion Asymptotics for Sine\b{eta} Correlation Functions

Abstract

Recent work gives an all-$\beta$, all-order stochastic-zeta representation of the correlation functions of the $\Sine_\beta$ process and determines their leading Vandermonde asymptotics when several variables merge. We compute the first nontrivial correction throughout the regime $m\beta>1$. If $a_1,\ldots,a_m$ are distinct real numbers and \[ V(a)=\sum_{1\le i 1/2$. The proof combines a finite-$N$ rotational Ward identity, exact Hua--Pickrell trace moments, compact moment bounds for the stochastic-zeta entire function and its derivatives, and a quantitative multivariate expectation--Taylor lemma. As a by-product we evaluate \[ \E_{\HP_{\beta,m\beta/2}}\sum_x x^{-2}=\frac{m\beta}{4(m\beta-1)(2m+1)}. \] The pole at $m\beta=1$ marks the boundary of the present second-moment argument and suggests a transition in the form of the next fusion correction.